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Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation

Louis Garénaux, Björn de Rijk

TL;DR

The paper analyzes the long-time behavior of small-amplitude solutions to the viscoelastic Klein-Gordon equation on ℝ with damping, establishing global existence and diffusive decay under a sign condition on the leading nonlinear terms. It develops a space-time resonances framework for a dissipative system, employing mode filtering and a near-identity transform to eliminate nonresonant quadratic and cubic terms, and derives a reduced amplitude equation for a leading Gaussian mode. The sign of the resonant cubic coefficient dictates whether the nonlinear interaction is absorption-like (global decay) or potentially destabilizing on exponential time scales. Consequently, the equilibrium u≡0 is nonlinearly stable for small localized perturbations, with explicit diffusive decay rates, and the analysis extends the space-time resonance method to dissipative wave equations. When the sign condition fails, the same approach yields diffusion up to an exponentially long time, after which the resonant term can drive slower dynamics governed by a separable ODE.

Abstract

We investigate the long-time behavior of solutions with small initial data to the viscoelastic Klein-Gordon equation with general smooth nonlinearity. Our analysis relies on the space-time resonances method to eliminate all nonresonant quadratic and cubic terms. We identify a sign condition for the remaining critical resonant term to be of absorption type, leading to global-in-time existence and diffusive decay of solutions with small initial data. Even when this condition fails, our analysis shows existence and diffusive decay of small solutions on exponentially long time intervals.

Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation

TL;DR

The paper analyzes the long-time behavior of small-amplitude solutions to the viscoelastic Klein-Gordon equation on ℝ with damping, establishing global existence and diffusive decay under a sign condition on the leading nonlinear terms. It develops a space-time resonances framework for a dissipative system, employing mode filtering and a near-identity transform to eliminate nonresonant quadratic and cubic terms, and derives a reduced amplitude equation for a leading Gaussian mode. The sign of the resonant cubic coefficient dictates whether the nonlinear interaction is absorption-like (global decay) or potentially destabilizing on exponential time scales. Consequently, the equilibrium u≡0 is nonlinearly stable for small localized perturbations, with explicit diffusive decay rates, and the analysis extends the space-time resonance method to dissipative wave equations. When the sign condition fails, the same approach yields diffusion up to an exponentially long time, after which the resonant term can drive slower dynamics governed by a separable ODE.

Abstract

We investigate the long-time behavior of solutions with small initial data to the viscoelastic Klein-Gordon equation with general smooth nonlinearity. Our analysis relies on the space-time resonances method to eliminate all nonresonant quadratic and cubic terms. We identify a sign condition for the remaining critical resonant term to be of absorption type, leading to global-in-time existence and diffusive decay of solutions with small initial data. Even when this condition fails, our analysis shows existence and diffusive decay of small solutions on exponentially long time intervals.
Paper Structure (14 sections, 12 theorems, 277 equations, 2 figures)

This paper contains 14 sections, 12 theorems, 277 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Technical summary
  4. Organization.
  5. Notation and function spaces
  6. Local existence and uniqueness
  7. Linear estimates
  8. Mode filters
  9. Separating relevant from irrelevant nonlinear terms
  10. Eliminating quadratic terms
  11. Eliminating nonresonant cubic terms
  12. Resonant cubic terms
  13. Reduced equations
  14. Proof of main results

Key Result

Theorem 1.1

Let $\alpha > 0$. Take $N \in C^4(\mathbb{R})$ such that $N(0) = 0$, $N'(0) = 0$, and the inequality e:signcondition holds. Then, there exist positive constants $M_0$ and $\varepsilon$ such that, whenever $u_0 \in H^2(\mathbb{R})$ and $w_0 \in L^2(\mathbb{R})$ satisfy $\hat{u}_0,\hat{w}_0 \in W^{1,\ there exists a unique global classical solution of the viscoelastic Klein-Gordon equation e:vKG2 w

Figures (2)

  • Figure 1: Depiction of the spectrum of the operator $\Lambda$, which consists of the half line ${(-\infty,-\tfrac{1}{\alpha}]}$ and the intersection of the closed left-half plane with the circle with center ${-\tfrac{1}{\alpha}}$ and radius ${\sqrt{1+\alpha^{-2}}}$. At frequency $k = 0$, the curves $\lambda_\pm(k)$ (depicted in red and blue) touch the imaginary axis in a quadratic tangency at the points $\pm \mathrm{i}$.
  • Figure 2: Depiction of the set \ref{['e:speccurve2']} for $0 < \gamma < 2$ (left panel) and for $\gamma > 2$ (right panel) under the condition that $\alpha^2 - \alpha \gamma + 1 > 0$. In case $\alpha^2 - \alpha \gamma + 1 \leq 0$, the set \ref{['e:speccurve2']} is confined to the negative real line.

Theorems & Definitions (28)

  • Theorem 1.1: Global existence and diffusive decay
  • Theorem 1.2: Existence and diffusive decay on exponentially long time scales
  • Remark 1.3
  • Remark 1.4
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 4.1
  • ...and 18 more